Calculus

Dale Varberg, Edwin Purcell, Steve Rigdon

Chapter 5

Applications of the Integral - all with Video Answers

Educators

Section 1

The Area of a Plane Region

Problem 1

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

James Kiss

Numerade Educator

Problem 1

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 2

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 3

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 4

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 5

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 6

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 7

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 9

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 10

In Problems $1-10,$ use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.

James Kiss

Numerade Educator

Problem 11

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=3-\frac{1}{3} x^{2}, y=0, \text { between } x=0 \text { and } x=3 $$

Monica Miller

Numerade Educator

Problem 12

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=5 x-x^{2}, y=0, \text { between } x=1 \text { and } x=3 $$

Monica Miller

Numerade Educator

Problem 13

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=(x-4)(x+2), y=0, \text { between } x=0 \text { and } x=3
= $$

Monica Miller

Numerade Educator

Problem 14

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=x^{2}-4 x-5, y=0, \text { between } x=-1 \text { and } x=4 $$

Monica Miller

Numerade Educator

Problem 15

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=\frac{1}{4}\left(x^{2}-7\right), y=0, \text { between } x=0 \text { and } x=2 $$

Monica Miller

Numerade Educator

Problem 16

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=x^{3}, y=0, \text { between } x=-3 \text { and } x=3 $$

Monica Miller

Numerade Educator

Problem 17

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=\sqrt[3]{x}, y=0, \text { between } x=-2 \text { and } x=2 $$

Monica Miller

Numerade Educator

Problem 18

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=\sqrt{x}-10, y=0, \text { between } x=0 \text { and } x=9 $$

Monica Miller

Numerade Educator

Problem 19

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=(x-3)(x-1), y=x $$

Nicole Wood

Numerade Educator

Problem 20

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=\sqrt{x}, y=x-4, x=0 $$

Monica Miller

Numerade Educator

Problem 21

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=x^{2}-2 x, y=-x^{2} $$

Monica Miller

Numerade Educator

Problem 22

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ y=x^{2}-9, y=(2 x-1)(x+3) $$

Monica Miller

Numerade Educator

Problem 23

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ x=8 y-y^{2}, x=0 $$

Monica Miller

Numerade Educator

Problem 24

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ x=(3-y)(y+1), x=0 $$

Monica Miller

Numerade Educator

Problem 25

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ x=-6 y^{2}+4 y, x+3 y-2=0 $$

Monica Miller

Numerade Educator

Problem 26

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ x=y^{2}-2 y, x-y-4=0 $$

Monica Miller

Numerade Educator

Problem 27

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ 4 y^{2}-2 x=0,4 y^{2}+4 x-12=0 $$

Monica Miller

Numerade Educator

Problem 28

$\approx$ In Problems $11-28$, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$$ x=4 y^{4}, x=8-4 y^{4} $$

Monica Miller

Numerade Educator

Problem 29

Sketch the region $R$ bounded by $y=x+6, y=x^{3},$ and $2 y+x=0 .$ Then find its area. Hint: Divide $R$ into two pieces.

Monica Miller

Numerade Educator

Problem 30

Find the area of the triangle with vertices at (-1,4) , $(2,-2),$ and (5,1) by integration.

Monica Miller

Numerade Educator

Problem 31

An object moves along a line so that its velocity at time $t$ is $v(t)=3 t^{2}-24 t+36$ feet per second. Find the displacement and total distance traveled by the object for $-1 \leq t \leq 9$.

Monica Miller

Numerade Educator

Problem 32

Follow the directions of Problem 31 if $v(t)=\frac{1}{2}+\sin 2 t$ and the interval is $0 \leq t \leq 3 \pi / 2$.

Monica Miller

Numerade Educator

Problem 33

Starting at $s=0$ when $t=0$, an object moves along a line so that its velocity at time $t$ is $v(t)=2 t-4$ centimeters per second. How long will it take to get to $s=12 ?$ To travel a total distance of 12 centimeters?

Monica Miller

Numerade Educator

Problem 34

Consider the curve $y=1 / x^{2}$ for $1 \leq x \leq 6$.
(a) Calculate the area under this curve.
(b) Determine $c$ so that the line $x=c$ bisects the area of part (a).
(c) Determine $d$ so that the line $y=d$ bisects the area of part (a).?

Nicole Wood

Numerade Educator

Problem 35

Calculate areas $A, B, C,$ and $D$ in Figure $12 .$ Check by calculating $A+B+C+D$ in one integration.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 36

Prove Cavalieri's Principle. (Bonaventura Cavalieri $(1598-1647)$ developed this principle in $1635 .$ ) If two regions have the same height at every $x$ in $[a, b]$, then they have the same area (see Figure 13$)$.

Xiaomeng Zhang

Numerade Educator

Problem 37

Use Cavalieri's Principle (not integration; see Problem 36) to show that the shaded regions in Figure 14 have the same area.

R M

Numerade Educator

Problem 38

Find the area of the region trapped between $y=\sin x$ and $y=\frac{1}{2}, 0 \leq x \leq 17 \pi / 6$.

Monica Miller

Numerade Educator